| import OAI.NumberTheory.Jacobsthal.Paths.GeometricRegularWords |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
|
|
| namespace NumberTheoryLean.FirstIsolatedPosition |
| open PrimeHistories GeometricRegularWords IsolatedBinGeometry |
| open LogarithmicBinScale LogarithmicBinLabels |
|
|
|
|
| theorem first_matching_position {α : Type*} (P : α → Prop) (ps : List α) |
| (hex : ∃ p ∈ ps,P p) : ∃ pre p tail,ps=pre++p::tail ∧ P p ∧ ∀ q ∈ pre,¬P q := by |
| classical |
| induction ps with |
| | nil => simp at hex |
| | cons p ps ih => |
| by_cases hp : P p |
| · exact ⟨[],p,ps,rfl,hp,by simp⟩ |
| · have ht : ∃ q ∈ ps,P q := by |
| obtain ⟨q,hq,hP⟩ := hex |
| rcases List.mem_cons.mp hq with rfl | hq |
| · exact False.elim (hp hP) |
| · exact ⟨q,hq,hP⟩ |
| obtain ⟨pre,q,tail,he,hq,hpre⟩ := ih ht |
| exact ⟨p::pre,q,tail,by rw [List.cons_append,he],hq,by simpa using And.intro hp hpre⟩ |
|
|
| theorem geometric_first_isolated {w top xi C B R L alpha beta : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (z : Node) (ps : List ℕ) |
| (hp : ps ∈ geometricWords hw htop hxi C B R L alpha beta z) : |
| ∃ pre p tail,ps=pre++p::tail ∧ |
| isolatedBin w top xi B alpha beta (label (zero_lt_one.trans hw) htop hxi p) ∧ |
| ∀ q ∈ pre,¬isolatedBin w top xi B alpha beta (label (zero_lt_one.trans hw) htop hxi q) := |
| first_matching_position _ ps (geometric_isolated_prime hw htop hxi z ps hp) |
| end NumberTheoryLean.FirstIsolatedPosition |
|
|
|
|
| end Erdos970 |
|
|
| end OAI |
|
|
